Quantum geometric tensor away from equilibrium

Abstract

Keywords: quantum many-body theory, quantum systems away from equilibrium, geometry of quantum states, quantum phase transitions, quantum adiabaticity, information geometry AbstractThe manifold of ground states of a family of quantum Hamiltonians can be endowed with a quantum geometric tensor whose singularities signal quantum phase transitions and give a general way to define quantum phases. In this paper, we show that the same information-theoretic and geometrical approach can be used to describe the geometry of quantum states away from equilibrium. We construct the quantum geometric tensor Q μν for ensembles of states that evolve in time and study its phase diagram and equilibration properties. If the initial ensemble is the manifold of ground states, we show that the phase diagram is conserved, that the geometric tensor equilibrates after a quantum quench, and that its time behavior is governed by out-of-time-order commutators (OTOCs). We finally demonstrate our results in the exactly solvable Cluster-XY model.

Towards a Geometrization of Quantum Complexity and Chaos

“…From this bound some important results descend, as the existence of a finite correlation length χ for non-degenerate ground states [20]. Generalizing these results it is possible to prove [11] the following:…”

“…To extend this result to the QGT on M t , we show that equilibration in probability also applies to the correlation functions of local operators. In particular the following holds [11] Theorem 1. Consider a Hamiltonian H q = E q n P q n satisfying the non-resonance condition and an observable A(t) = e −itH q Ae itH q evolving under the action of H q .…”

“…Taking advantage of the fact that for any t ≥ 0 M t is a manifold of nondegenerate ground states for the family of Hamiltonians H(λ) t := U t (λ)H(λ)U t (λ) † , one can prove [11] that, in a coordinate map a i (ψ) which diagonalizes the tensor, the eigenvalues q a of the QGT are bounded as follows…”

“…To represent q 1 as the sum of local correlations, we exploit the locality of the quench Hamiltonian. We can indeed represent the operator D as D = i D i , where [11]. As a consequence…”

“…In this paper, we demonstrate a formalism to describe the time evolution of quantum geometry [11] defining a protocol to study the evolution of the Fubini-Study geometry on the manifold of ground states of a smooth family of local Hamiltonians, that is, the out-of-equilibrium evolution of the phase diagram. We show that quantum phases are robust after a quantum quench and the geometry of the manifold of evolving states is affected by a process of equilibration.…”